Time varying (TV) wireless channel models capture both the space and time variations of wireless systems, which are due to the relative mobility of the receiver, transmitter and/or scatterers [1] -[4] . The majority of re- search papers in this field such as in [5] -[7] use time-invariant (static) models for wireless channels. In these models, the speeds of mobile nodes are assumed to be constant and the statistical characteristics of the received signal are assumed to be fixed in time. But in reality, the propagation environment varies continuously due to mobility of the nodes causing network topology to dynamically change, the angle of arrival of the wave upon the receiver can vary continuously, and objects or scatters move in between the transmitter and the receiver resulting in appearance or disappearance of existing paths from one instant to the next. As a result, the current models that assume fixed statistics can no longer capture and track complex time variations in the propagation environment. These time variations compel us to introduce more advanced dynamical models based on stochastic state-space representation, in order to capture higher order dynamics of the wireless channel. In time-invariant models, channel parameters are random but do not depend on time, and remain constant throughout the observation and estimation phase. This contrasts with TV models, where the channel dynamics become stochastic processes [1] - [4] . This paper focuses on the development of stochastic short-term fading channel models based on system identification algorithms using received signal measurement data to extract various channel parameters and ap- ply an optimal power control scheme.

In [4] , the TV channel parameters are estimated from approximating the Doppler power spectral density (DPSD) of the wireless fading channel. However, in reality one does not have access to the TV DPSD at all times during the estimation process. Since these models are based on state-space forms, we propose to estimate the channel parameters as well as the inphase and quadrature components directly from received signal mea- surements, which are usually available or easy to obtain in any wireless network. A filter-based expectation maximization (EM) algorithm [8] and Kalman filter [9] are employed in estimating the channel parameters as well as the inphase and quadrature components, respectively. These recursive filters use only the first and second order statistics and therefore can be implemented online. These algorithms have been recently utilized in [1] and [2] to estimate the wireless channel parameters and states, and therefore the formulations of these algorithms are not presented in this paper. The proposed models and estimation algorithms are tested using received signal strength measurement data and the results are presented.

The developed stochastic channel models from received signal strength measurements are useful in most wireless applications. In this paper, these models are used to develop an optimal power control algorithm (PCA). Power control (PC) is important to improve performance of wireless communication systems. The benefits of power minimization are not just increased battery life, but also increased overall network capacity. The power allocation problem has been studied extensively as an eigenvalue problem for non-negative matrices [10] [11] , resulting in iterative PCAs that converge each user’s power to the minimum power [12] [13] , and as optimiza- tion-based approaches [14] . Much of this previous work deals with static time-invariant channel models.

The proposed PCA is based on predictable power control strategies (PPCS) that were first introduced in [4] . PPCS basically means updating the transmitted powers at discrete times and maintaining them fixed until the next power update begins. The PPCS mechanism is proven to be effectively applicable to such dynamical mod- els for an optimal PC. The outage probability (OP) is used as a performance measure. Since few TV dynamical channel models have so far been investigated with the application of any PCA, the suggested dynamical models and PCA thus provide a far more realistic and efficient optimal control of wireless networks.

The remainder of this paper is organized as follows. In Section 2, stochastic modeling and online estimation of TV wireless fading communication channels are presented. Section 3 introduces an optimal PCA based on predictable strategies and employing the estimated channel parameters and states. Section 4 presents numerical results that validate the proposed models together with the estimation and power control algorithms. Finally, Section 5 provides concluding remarks.

The general TV model of a wireless channel is typically represented by the following multipath band-pass im- pulse response [6]

where is the band-pass response of the channel at time, due to an impulse applied at time, is the random number of multipath components, is the carrier frequency, is the Dirac delta

function, and the set describes the random TV inphase component, quadrature

component, and arrival time of the different paths, respectively. Let be the low-pass equivalent represen- tation of the transmitted signal, then the band-pass representation of the received signal is given by

where and are two independent and identically distributed (iid) white Gaussian noise processes.

It is shown in [4] and [15] that the DPSD of a wireless fading channel, denoted by, can be approxi- mated by an even, stable, rational, and factorizable transfer function, , where is given by

Consequently, the inphase and quadrature components can be realized using the following stochastic state- space representation [16]

where

and are the state vectors corresponding to the inphase and quadrature components, re-

spectively., are independent standard Brownian motions, which correspond to the in-

phase and quadrature components of the path respectively, and are arbitrary functions representing the presence or absence of line-of-sight (LOS) of the inphase and quadrature components for the path respectively, and denotes matrix or vector transpose. Without loss of generality, we consider the case of flat fading, in which the fading channel has purely a multiplicative effect on the signal and the multipath components are not resolvable, and, thus, can be considered as a single path [6] . We also consider the non-line- of-sight (NLOS) case, i.e., , which represents an environment with large obstructions. The time-varying state space model described in (4) has a solution given by

where or, is the state transition matrix associated to, and . Therefore, the mean of is

and the covariance matrix of is

For the time-invariant case, , then (6), (7), and (8) sim- plify to

It can be observed in (7) and (8) that the mean and variance of the inphase and quadrature components are functions of time. Thus, the statistics of the inphase and quadrature components, and therefore the statistics of the channel, are times varying.

Similarly, following the state space representation in (4) and the received signal in (2), the fading channel can be represented using general stochastic state-space representation of the form

where

In this case, represents the received signal measurements, is the state variable of the inphase and quadrature components, and is the measurement Gaussian noise.

In [4] and [15] , the channel parameters are obtained from approximat-

ing the DPSD. However, in reality one does not have access to the DPSD at all times during the estimation process. Therefore, we propose estimating the channel parameters as well asinphase and quadrature components directly from received signal measurements, which are usually available or easy to obtain in any wireless net- work. A filter-based expectation maximization (EM) algorithm combined with the Kalman filter is employed to estimate the channel model parameters and states in (10). These filters use only the first and second order statis- tics and are also recursive, and therefore can be implemented online. These algorithms have been recently uti- lized in [1] and [2] to estimate wireless channel parameters and states, and therefore the formulations of these algorithms are not presented in this paper. Experimental results demonstrating the applicability of these algo- rithms in conjunction with the proposed stochastic wireless models are discussed in Section 4. In the next section, we introduce an important application based on the developed models; that is stochastic PC in wireless networks.

In this section, an optimal PCA is investigated based on the estimated wireless channel models. Since the chan- nel model parameters are estimated from received signal measurements, PC can be performed solely from hav- ing these measurements. The aim of the PCA described here is to minimize the total transmitted power of all us- ers while maintaining acceptable quality-of-service (QoS) for each user. The measure of QoS is defined by the signal-to-interference ratio (SIR) for each link to be larger than a target SIR.

By generalizing the wireless channel model in (10), the state-space representation of a wireless network with transmitters and receivers can be described as

where is the received signal at the receiver at time, is the states of the channel between transmitter and the receiver assigned to transmitter, is the transmitted power of transmitter at time, which acts as a scaling on the information signal, is the channel disturbance or noise at receiver, and.

Consider the wireless network described in (12), the centralized PC problem for TV channels over a time in- terval can be stated as follows [4]

where is the target SIR at receiver and. A solution to (13) is presented by first introducing the communication meaning of predictable power control strategies (PPCS). In wireless cellular networks, it is practical to observe and estimate channels at the base stations and then send the information back to the mobiles

to adjust their power signals. Since channels experience delays, and power control is not feasible continuously in time but only at discrete time instants, the concept of predictable strategies is introduced [4] . Consider a set of discrete time strategies,. At time, the base stations estimate the channel information as illustrated in Section 2. Using the concept of predictable strategy, the base stations determine the control strategy for the next time in- stant. The latter is communicated back to the mobiles, which hold these values during the time interval. At time, a new set of channel information is estimated at base sta- tions and the time control strategies are computed and communicated back to the mobiles which hold them constant during the time interval. Such decision strategies are called predictable. Us- ing the concept of PPCS over any time interval, Expression (13) is equivalent to

where

and denotes a diagonal matrix with its argument as diagonal entries. The optimization in (14) is a li-

near programming problem in vector of unknowns. Here is a time interval such that the channel model does not change significantly, i.e., should be smaller than the coherence time of the channel. Throughout this section, we assume that the PC problem is feasible, i.e., there exists a power vector that satisfies the inequality in (14) for all in.

In the next section, a numerical example is presented to determine the performance of the proposed PCA un- der the estimated wireless channel models.

In this section, two numerical examples are presented. In Example 1, the EM algorithm combined with Kalman filtering is performed to estimate the channel parameters as well as inphase and quadrature components from received signal measurements. In Example 2, the performance of the proposed PCA based on the estimated channel models in Example 1 is determined and compared with fixed transmitted powers.

This paper describes a general scheme for extracting mathematical wireless channel models from noisy received signal measurements, and performing power control based on the estimated channel parameters. The channel models are represented in stochastic state-space form. The proposed estimation algorithm consists of filtering based on the Kalman filter to remove noise from data, and identification based on the EM algorithm to deter- mine the parameters of the model which best describe the measurements. Numerical results indicate that the measured data can be regenerated through a simple order discrete-time stochastic state-space model. More- over, a stochastic PCA based on the estimated parameters and channel states is investigated. Numerical results indicate that there is potentially large gain to be achieved by using the proposed PCA, which can be used as long as the channel model does not change significantly; that is is a subset of the coherence time of the channel.

This paper has been authored by employees of UT-Battelle, LLC, under contract DE-AC05-00OR22725 with the US Department of Energy. This work was also supported in part by NSF grant CMMI-1334094.